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I'm working on some data where I have thousands of curves in the defined as: $P(t)$ that basically represent eluition profiles of some proteins.

My aim is to represent each of this $P(t)$ curves as a linear combination of some other curves plus some error. More specifically:

$P(t) = \beta_{1} M(t) + \beta_2L(t)+\beta_3Z(t)+\beta_4G(t)+\epsilon$

Basically I want to find the scalar coefficient $\beta_1,\beta_2,..,\beta_n$ that minimize the residual sum of squares on $P(t)$ for me it seems like a simple multiple regression problem. The only issue seems to be that we are talking about functional data.

It seems that maybe can be a very simple problem, but I don't really know if I can just use simple linear regression to estimate the coefficients. I've heard about functional linear regression that takes the forms:

$P(t) = \beta_0(t)+\beta_1(t)M(t)+\beta_2(t)L(t)+\epsilon$

But in this case the coefficients are defined as functions, I really need them to be expressed as a scalar. Maybe I'm just a little bit confused, but It seems a very simple problem, there is a feasible way to resolve it?

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  • $\begingroup$ Do you have a preferred programming language? $\endgroup$ Jul 14, 2018 at 12:43
  • $\begingroup$ Yes, I use mainly R. It really seems a simple problem, but I'm not sure if I can resolve it with the classical lm() function. Unfortunately I think no. $\endgroup$ Jul 14, 2018 at 13:31
  • $\begingroup$ Any hints or suggestions? :) $\endgroup$ Jul 15, 2018 at 12:04
  • $\begingroup$ For y = f(x): if I have three vectors where the first vector is all 1.0, the second contains the values of x, and the third contains the values of x^2, when I perform linear regression with linear algebra of y against these vectors I will have the coefficients a, b, c for the equation "y = a + bx + cx^2". Might you do something similar in this case, where you create vectors of the function values and then use standard linear algebra? $\endgroup$ Jul 15, 2018 at 13:04

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Both models you mention fit into the framework of the varying coefficient models described originally by Hastie and Tibshirani (1993).

From a functional data perspective, the second model you reference is called a functional current regression model. The first model is just a special case of the concurrent model where the coefficient functions are constant. For references, see chapter 14 of Ramsay and Silverman's book (2005) or, more recently, Leroux et. al. (2018). As with most functional data methods, whether your functional data are sparse or dense affects how you should conduct your analysis; the latter reference details a model-fitting procedure for when the functional data are sparsely sampled.

If you would like to fit such a model using R, the fcr package was designed to accompany the paper by Leroux et. al. (2018), but the pffr function of the refund package can also be used to accomplish your goal.

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